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Charge-carrier states and light absorption in ordered quantum dot superlattices
D. L. Nika,* E. P. Pokatilov,* Q. Shao, and A. A. Balandin†
Nano-Device Laboratory, Department of Electrical Engineering, University of California-Riverside, Riverside, California 92521, USA�Received 20 May 2007; published 14 September 2007�
We have theoretically investigated the electron and hole energy spectra and light absorption in the three-dimensionally ordered quantum dots superlattices �QDS� made of the direct band-gap semiconductors. Thecalculations were performed for QDS of the rhombic symmetry with a substantial electron �hole� wave-function overlap using a one-band Hamiltonian for the electrons and six-band Hamiltonian for the holes. Theobtained results were compared with the predictions of the simplified models for the uncoupled heavy, light,and split-off holes. It has been shown that the energy spectra of the electrons and holes in the ordered QDS aredistinctively different from those in the single quantum dots �QD� or conventional quantum-well superlattices.The charge-carrier dispersion and localization are very sensitive to the quasicrystallographic directions definedby the dots, which play a role of the atoms in such QD supracrystal. We found that in the ordered QDS theoscillator strength for the interband optical transitions can be high for a relatively wide range of the photonenergies. The obtained results are important for the proposed applications of QDS in solar cells, photodetectors,and thermoelectrics.
DOI: 10.1103/PhysRevB.76.125417 PACS number�s�: 73.21.�b, 73.22.�f, 73.21.La
I. INTRODUCTION
Nanostructures, such as multiple arrays of quantum dots�QD�, have recently attracted significant attention owing totheir unusual physical properties and proposed applicationsin photovoltaic solar cells,1,2 photodetectors,3–5 thermoelec-tric elements6–8 as well as electronic circuits based on thealternative state variables.9,10 Some of these applications canbe implemented with the arrays of the uncoupled �or weaklycoupled� and randomly distributed quantum dots—others re-quire the long-range ordering of the quantum dots or precisecontrol of the quantum dot position. While the properties ofthe single quantum dots made of different material systemshave been extensively investigated theoretically,11,12 only alimited number of studies addressed the ensemble propertiesof the multiple arrays of quantum dots with a significantelectron �or hole� wave-function overlap.13,14
Three-dimensional �3D� ordering of the closely spacedQD in the quantum dot superlattices �QDS� results in theformation of the carrier minibands. The latter was demon-strated experimentally in a range of systems.15,16 Lazaren-kova and Balandin13 reported a theoretical study where theycalculated the electron states in the Kronig-Penny-type ap-proach using a model confining potential, which led to thewave-function separation. Although their model is conve-nient for a simple calculation of the electronic states, it ishard to predict how closely the model potential replicates therealistic band offsets, particularly for the higher-energy mini-bands. In this paper we propose a rigorous approach for de-termining the electron and hole states in 3D ordered QDSand calculating the oscillator strength for the optical transi-tions in such structures. The proposed theoretical model andcalculation procedure can be used for the prediction of thecarrier transport and light absorption in the ordered QDSmade of the direct band-gap semiconductors. There is astrong practical need in such a model taking into account thelarge scale of the experimental efforts focused on applica-tions of QDS in photovoltaic cells, lasers, photodetectors,and thermoelectric elements.
The calculations, presented in this work, are carried outfor InAs QD in a InGaAs host matrix. The selected materialsystem is particularly interesting for a theoretical study be-cause it was in the focus of many recent experimentalinvestigations.17–25 In the next section we present our theo-retical formalism for determining the electron and hole en-ergy dispersion in the ordered QDS. The hole states aretreated using the six-band Hamiltonian, which allows us totake into account the interactions among the heavy, light, andspin-off hole bands. In Sec. III we report a comparison of ourresults with the prediction of the Lazarenkova andBalandin13 model for the electron states in QDS with thespecial wave-function separable potential. The calculationof the oscillator strength of the optical transitions in 3D or-dered QDS are reported in Sec. IV. We give our conclusionsin Sec. V.
II. ELECTRON AND HOLE STATES IN THE ORDEREDQUANTUM DOT SUPERLATTICES
We consider a generic QDS with the dots ordered in allthree dimensions. Although a fabrication of such 3D orderedQDS is still a major technological challenge, there have beena number of reports of achieving a nearly perfect 3D order-ing in QDS.26 It is expected that further progress in the self-assemble growth will result in a greater availability of theordered QDS structures. Since we are mostly interested inthe ensemble properties of QDS and the features, whichoriginate from the dot-dot interaction �through the wave-function penetration�, we assume the simplest orthorhombicsymmetry for the dots with a corresponding rectangularprism unit cell �see the schematic in Fig. 1�. The space sym-metry group of such lattice is D2h
24. In this structure, InAsQDs are located in the nodal points of the lattice filled withGa0.5In0.5As, which acts as the barrier material. The dimen-sions of the QD sides are denoted by l1, l2, l3 �where indices1, 2, and 3 indicate axis X, Y, and Z of the Cartesian coor-dinate system, respectively�. The interdot distances �thick-
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ness of the barriers� are denoted by h1, h2, and h3. The trans-lation periods of the lattice are di= li+hi�i=1,2 ,3� with thelattice basis vectors a�1�a1 ,0 ,0�, a�2�0,a2 ,0�, and a�3�0,0 ,a3�,where ai=di �i=1,2 ,3�.
The basis vectors of the reciprocal lattice, required for theconstruction of the quasi Brillouin zone �QBZ�, are
b�1�b1 ,0 ,0�, b�2�0,b2 ,0�, and b�3�0,0 ,b3�, where bi=2� /di �i=1,2 ,3�. QBZ in QDS supracrystal, i.e., 3D ordered QDS, isanalogous to the conventional Brillouin zone for crystal sol-ids, which is defined as a Wigner-Seitz cell of the reciprocallattice. In the QDS supracrystal, the quantum dots play a roleof the atoms. In the atomic crystal lattices the exact wavefunctions of the charge carriers cannot be obtained due to thedifficulties of finding the actual periodical lattice potentials.In QDS the periodic potential is well defined and the solu-tions of the Schrödinger equation can be presented explicitly.In our analysis we consider the electron and hole states sepa-rately due to a sufficiently large energy gap between theelectron and hole states in QD, which is increased by the sizequantization and reaches a value �0.64 eV for the consid-ered dimensions. Owing to the translational symmetry in theordered QDS, the electron �hole� envelope wave functions,defined as in the effective mass approximation, obeys therelation
�ne,h�r� + a�� = �n
e,h�r��eik�a� . �1�
Here k� is the quasicontinuous wave vector of the electron�hole�, n is the quantum number of the size-quantized elec-tron �hole� states, a� =N1a�1+N2a�2+N3a�3 is the lattice vector,where N1, N2, and N3 are the arbitrary integer numbers. Themaximum value of k� is restricted by the QBZ boundaries.
The one-band Schrödinger equation for an electron inQDS can be written as
�−�2
2�
1
me�r��� + Ve�r����n
e�r�� = Ene�n
e�r�� , �2�
where Ve�r�� is the potential energy, me�r�� is the electron ef-fective mass, and � is the Plank constant. The potential en-ergy Ve�r��=0 and the electron mass me�r��=me�InAs�
=0.026m0 inside the QD boundaries defined as �xi− �a��i ��
li
2 �the coordinate origin is chosen in the center of QD�.Outside of the QD boundaries �inside the barrier layers� de-fined as
li
2 � �xi− �a��i�, the following values of the potentialand mass are assumed: Ve�r��=VB
e =const and me�r��=me�Ga0.5In0.5As�=0.0435 m0, where m0 is the electronmass in vacuum. Eq. �2� with the periodic boundary condi-tions given by Eq. �1� was solved numerically using the finitedifference method.
The electron energy dispersions, calculated for three dif-ferent QDS along three quasicrystallographic directions, arepresented in Fig. 2. The results are shown for lx=5 nm, ly
=5.5 nm, lz=6 nm, and k� = �k ,0 ,0� �panel �a��; lx=5 nm, ly
=5 nm, lz=6 nm, and k� = �k ,k ,0� �panel �b��; and lx=5 nm,ly =5 nm, lz=5 nm, and k� = �k ,k ,k� �panel �c��. The barrierthicknesses are hx=hy =hz=2 nm for all of the consideredQDS. The maximum values of the electron wave vectork used for normalization in Fig. 2 are kmax�1,0 ,0�=0.448 nm−1, kmax�1,1 ,0�=0.663 nm−1, and kmax�1,1 ,1�
Y
Z
LxHx
Dx
XHy Ly Dy
Lz
Hz
Dz
FIG. 1. �Color online� Schematic of the three-dimensionally or-dered quantum dot superlattice showing notations for the quantumdot sizes and interdot spacing.
k/kmax(1,0,0)
( ,0,0)k
(a)
VB
e
Ele
ctro
nE
ner
gy
(eV
)
( , ,0)k k
(b)
VB
e
Ele
ctro
nE
ner
gy
(eV
)
k/kmax(1,1,0)
VB
e
( , , )k k k
(c)
Ele
ctro
nE
ner
gy
(eV
)
k/kmax(1,1,1)
FIG. 2. Electron dispersion in InAs/Ga0.5In0.5As QDS �a� alongthe �1,0,0� quasicrystallographic direction with the QDS dimensionslx=5.0 nm, ly =5.5 nm, lz=6.0 nm, and hx=hy =hz=2.0 nm; �b�along �1,1,0� with the dimensions lx= ly =5.0 nm, lz=6.0 nm, andhx=hy =hz=2.0 nm; �c� along �1,1,1� with the dimensions lx= ly = lz
=5.0 nm and hx=hy =hz=2.0 nm. The electron energies are mea-sured from the conduction band bottom. The values of the potentialbarrier are shown in all figures by the straight line.
NIKA et al. PHYSICAL REVIEW B 76, 125417 �2007�
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=0.776 nm−1. The electron energies are counted from thebottom of the potential well.
One can see from Fig. 2 that the common features for allconsidered QDS include the strong size quantization of theelectron energy spectra and appearance of the relatively wideminibands for all n. The width of the lowest miniband is onthe order of 6.5kBT at room temperature, where kB is theBoltzman constant and T is an absolute temperature. This isa large value, which implies that the electrons move ratherfreely though the miniband. Another observation is that onlythe lowest �ground-state� miniband energies are below thepotential barrier level, i.e., ��k��VB
e �with the exception ofthe very large electron wave vectors k�. The value of thepotential barrier for the chosen system is VB
e
=0.83�Eg�Ga0.5In0.5As�-Eg�GaAs��=0.347 eV.27 The ener-gies of all excited minibands are larger than the potentialbarrier height VB
e . The electron energy dispersion for the ex-cited minibands differs strongly from the free-electron en-ergy dispersion or the near-parabolic dispersion in the con-duction band of bulk GaAs crystal. All dispersion branchesare double degenerate except for the first excited level shownin Fig. 2�c�, which is fourfold degenerate. The energy disper-sion in the ground state miniband is near parabolic for a wideinterval of k in the considered directions, i.e., �1,0,0�, �1,1,0�,and �1,1,1�. For comparison, in Fig. 2�c� we also show withthe dashed lines the energy dispersion for the ground mini-
band in ordered QDS with the cubic QD�lx= ly = lz=5 nm�along �1,0,0� and �1,1,0� quasicrystallographic directions. Itis clear from this figure that the equal-energy surfaces in kspace are nearly isotropic. Note that the lowest levels play amajor role in the kinetic processes at low and medium tem-peratures and moderate electron concentrations �character-ized by the Fermi energy smaller than 0.5 eV�. The non-monotonic behavior of the dispersion curves for all exitedminibands leads to a strong dependence of the electron ve-locity on the value of the electron wave vector k as well asappearance of the electron states with the negative velocities,which correspond to the region of the dispersion curves withnegative slopes.
The hole energies Enh and wave functions �n
h�r�� were ob-tained from the numerical solution of the Schrödinger equa-tion with the six-band Hamiltonian, which takes into accountthe interactions among the heavy, light, and spin-off holebands. The Hamiltonian is written as28,29
Hnh�� n
h = Enh�� n
h, �3�
where
Hh = HXYZ�r�h� 0
0 HXYZ�r�h� + HSO�r�h� , �4�
HXYZ�r�h� =�2
2m0�v� − kx�lkx − k�x��hk�x
� − 3�kx3+ky + ky3
−kx� − 3�kx3+kz + kz3
−kx�
− 3�kx3−ky + ky3
+kx� �v� − ky�lky − k�y��hk�y
� − 3�ky3+kz + kz3
−ky�
− 3�kx3−kz + kz3
+kx� − 3�ky3−kz + kz3
+ky� �v� − kz�lkz − k�z��hk�z
� , �5�
HSO�r�h� =�r�h�
3 0 − i 0 0 0 1
i 0 0 0 0 − i
0 0 0 − 1 i 0
0 0 − 1 0 i 0
0 0 − i − i 0 0
1 i 0 0 0 0
. �6�
In Eqs. �4�–�6�, we define �v�=�v−� /3, k� =−i�� , k�x,y,z� =k�
−k�x,y,z; �r�h�=�2��r�h� / �2m0� is the spin-orbit splittings ofthe valence band, Ev�r�h�=�2�v�r�h� / �2m0� is the energy of thevalence band maximum, and
�l = �l�r�h� = 1�r�h� + 42�r�h� ,
�h = �h�r�h� = 1�r�h� − 22�r�h� ,
3± = 3
±�r�h� = 3�r�h� ± ��r�h� ,
��r�h� = �22�r�h� + 33�r�h� − 1�r�h��/3. �7�
Inside the QD, the Luttinger parameters i�r�h�=i�InAs��i=1,2 ,3�, �r�h�=�InAs�, and Ev�r�h�=Ev�InAs�, whileoutside the QD �inside the barrier materials�i�r�h�=i�Ga0.5In0.5As�, �r�h�=�Ga0.5In0.5As�, and Ev�r�h�=Ev�Ga0.5In0.5As�. The material parameters required forthe calculation of the hole energy spectra were taken fromRef. 30.
The energy dispersion for the holes in the ordered QDS�with the same dimensions as those in Figs. 2� along thequasicrystallographic directions �1,0,0�, �1,1,0�, and �1,1,1�are shown in Figs. 3�a�–3�c�, respectively. The energy iscounted from the bottom of the valence band. ComparingFigs. 2 and 3, one can see that the quantization of the hole-energy spectra is weaker than that of the electron spectra.The ground state minibands in all considered cases have en-ergies smaller than 0.05 eV. The spin-orbit interaction splitsthe multifold degenerate hole-energy levels into the brancheswith the similar energies in the narrow region of a few meV.
CHARGE-CARRIER STATES AND LIGHT ABSORPTION IN… PHYSICAL REVIEW B 76, 125417 �2007�
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Along the �1,0,0� quasicrystallographic direction the hole en-ergies of the lowest branch are smaller than the valence bandbarrier energy EB
h =0.086 eV for all values of the hole wavevector k. Thus, the hole wave function of the split ground-state miniband is mostly located in the potential well. Alongthe �1,1,0� and �1,1,1� directions the hole energy rapidly in-
creases with increasing k and the holes become weakly lo-calized or nearly free states. Along the �1,0,0� and �1,1,0�directions the dispersion curves have the minimum andmaximum values whereas along the �1,1,1� direction the holeenergy increases monotonically with increasing of k �as inbulk crystals�. The revealed differences among the dispersioncurves along various directions are expected to manifestthemselves in the spatial anisotropy of the kinetic coeffi-cients in QDS.
Figure 3�d� demonstrates the strength of the hole stateintermixing by depicting the difference between the hole en-ergy spectra obtained using the six-band Hamiltonian, whichtakes into account the interactions between the hole bands,and the one-band Hamiltonians for the independent heavy,light, and spin-off holes. The heavy holes �indicated by solidlines� manifest small size-quantization �around 10 meV forthe exited states� and weak dispersion while the light andspin-off holes show large size quantization and strong disper-sion. All energy levels of these holes are twofold degenerate.Only the energies of the ground states of the heavy and thoseof the interacting “six-band” holes are similar. The indepen-dent systems of the weakly dispersive heavy holes andstrongly dispersive light and spin-off holes transform, in thecase of interacting “six-band” holes, into a system of the fourweakly dispersive lowest levels and strongly dispersivehigher levels with n 4. Thus, the properties of the light,spin-off, and heavy holes manifest themselves in each dis-persion curve of the interacting “six-band” holes for n 4.
III. COMPARISON WITH THE SEMIANALYTICALMODEL FOR THE QUANTUM DOT SUPERLATTICE
It is illustrative to compare the electronic dispersion ob-tained for the ordered QDS calculated using the approachoutlined in Sec. II with the prediction of the semianalyticalmodel,13 based on the approximate confining potential,which allows for the electron wave-function separation alongthe three coordinates. The model has been applied for thecalculation of the minibands in QDS-based solar cells.31 Inorder to perform the comparison, one must assume that theconfining potential is written as a sum of three independentperiodic functions of coordinates x, y, and z �with periods ofdx, dy, and dz�, i.e., V�r�=Vx�x�+Vy�y�+Vz�z�, where eachcomponent is given as
V���� = �0 if �� − ��d�� � L�/2,
V0 if �� − ��d�� L�/2.�8�
Here �� are the integer numbers and subscript � denotes aparticular coordinate axis. This choice of potential results innonuniform barrier height �larger potentials in the corner� butallows one to separate the carrier motion along three coordi-nate axes. As shown in Ref. 13, the solution of theSchrödinger equation for the QDS with the potential of Eq.�8� has the Kronig-Penny-type form and can be written as
cos�q�d�� = cos�k�WL��cos�k�
BH�� −1
2� k�
BmW*
k�WmB
* +k�
WmB*
k�BmW
* ��sin�k�
WL��sin�k�BH�� if E� � V0, �9a�
Hole
En
ergy
(eV
)
k/kmax(1,0,0)
( ,0,0)k
VB
h
Hole
En
ergy
(eV
)
( , , )k k k
(c)
k/kmax(1,1,1)
VB
h
k/kmax(1,1,0)
( , ,0)k k
VB
h
Hole
En
ergy
(eV
)
( ,0,0)k
k/kmax(1,0,0)
(d)
VB
h
lhhh
sh
Hole
En
ergy
(eV
)
(a)
(b)
FIG. 3. Hole dispersion obtained within the six-band model inInAs/Ga0.5In0.5As QDS �a� along the �1,0,0� quasicrystallographicdirection with the QDS dimensions lx=5.0 nm, ly =5.5 nm, lz
=6.0 nm, and hx=hy =hz=2.0 nm; �b� along �1,1,0� with the dimen-sions lx= ly =5.0 nm, lz=6.0 nm, and hx=hy =hz=2.0 nm; �c� along�1,1,1� with the dimensions lx= ly = lz=5.0 nm and hx=hy =hz
=2.0 nm. �d� The dispersion of heavy �solid lines�, light �dottedlines�, and spin-off �dashed lines� holes along �1,0,0� with the samedimensions as in �a�. The hole energies are measured from the topof the valence band and shown with the positive sign.
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cos�q�d�� = cos�k�WL��cosh�k�
BH�� −1
2�−
k�BmW
*
k�WmB
* +k�
WmB*
k�BmW
* ��sin�k�
WL��sinh�k�BH�� if 0 � E� � V0, �9b�
where
k�B =
�2mB* �E� − V0�
�, k�
W =�2mW
* �E���
. �10�
Here mB* and mW
* are the effective masses inside the barrierand quantum dot, correspondingly. One can calculate theelectron dispersion in the ordered QDS with this simplifyingpotential as E�q�=Ex�qx�+Ey�qy�+Ez�qz�.
Figure 4 shows the electron dispersion relation obtainedwith the model potential of Eq. �8�. The results are obtainedfor the ordered QDS with the same structure and materialparameters as those depicted in Fig. 2�a�. A direct compari-son of the dispersion relations in Fig. 4 and Fig. 2�a� indi-cates that the results agree well for the small energies �lowestminibands� but start to deviate for the above-the-barrierstates. The width of the lowest miniband is 5.8kBT at roomtemperature. This value is in good agreement with our cal-culations with an actual equal-height potential. Thus, it issafe to use the model potential and the Kronig-Penny-typesolution for the below-the-barrier states in the conductionband of the ordered QDS.
IV. ANALYSIS OF THE ELECTRONIC STATESIN THE ORDERED QUANTUM DOT SUPERLATTICES
In order to understand the nature of the electron and holestates in the ordered QDS and their differences from those inconventional quantum-well superlattices and bulk crystals,we calculated the electron �hole� density distributions in theordered QDS. The knowledge of the charge-carrier densitydistributions can help to elucidate the specifics of the charge-
carrier transport and light absorption by such artificial su-pracrystals. The electron density functions in the orderedQDS with the QD dimensions lx= ly = lz=5 nm and the barrierthicknesses hx=hy =hz=2 nm are shown in Figs. 5�a�, 5�b�,6�a�, and 6�b� for k=0 and k=kmax for the quasicrystallo-graphic directions �1,0,0�, �1,1,0�, and �1,1,1�.
Figure 5�a� shows the constant surface of the electronfunction ��n=1
e �2= cV0
, where V0=d1d2d3 is the volume of theunit cell of the ordered QDS and c=1.0. The cube plotted inFig. 5�a� represents the unit cell with the sides d1=d2=d3=7 nm. The potential quantum well with the sides lx= ly = lz=5 lies inside the cube. The surface ��n=1
e �r� ,k=0��2=const isthe invariant function of the QDS symmetry group. The re-gions of the high density of the electron wave function arerepresented by the spherical surfaces, while the regions ofthe lower density are represented by the channel-like sur-faces joined with the cube faces. The latter indicates that theelectron wave functions penetrate through the barriers in the
0.0 0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
1.2
VBe
(k,0,0)
ElectronEnergy(eV)
k/kmax(1,0,0)
FIG. 4. Electron dispersion obtained within the semianalyticalmodel which assumes that the confining potential can be presentedas V�r�=Vx�x�+Vy�y�+Vz�z�. The results are shown for QDS withthe same dimensions as in Fig. 3�a�.
( )a
( )b
FIG. 5. �Color online� Constant surface of the electron densityfunction ��n=1
e �2= cd1d2d3
in the ordered QDS with the dimensionslx= ly = lz=5 nm and hx=hy =hz=2 nm for �a� c=1.0 and the electronwave vector k=0.0; and �b� c=0.5 �outside� and c=2.0 �toroidalsurface� for k=kmax�1,0 ,0�.
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middle of the dot and periodically repeat themselves in allunit cells. The electron density function for k=kmax�1,0 ,0� ispresented in Fig. 5�b� for c=0.5 and c=2.0. A comparison ofFigs. 5�a� and 5�b� indicate that the electron density distribu-tion in QDS strongly depends on the value of the electronwave vector k. The high-density core of the electron function��n=1
e �2 is spherical for k=0 while it is ellipsoidal for k=kmax�1,0 ,0�. The channel-like surfaces along three direc-tions �at k=0� transform into the channel-like surfaces alongtwo directions �0,1,0� and �0,0,1�. The shape of the equal-density surfaces reflect the symmetry reduction introducedby the nonzero wave vector k. At k=kmax �1,0,0� the electrondensity is equal to zero in the middle of the barrier onthe �Y ,Z� plane and the electron energy reaches maximumvalue while the electron velocity becomes zero, i.e.,v�k= �(���k�)� ���k���max
=0.The dependence of the electron density function on k
along the �1,1,0� quasicrystallographic direction is somewhatanalogous. In this case, the electron density is compressed intwo directions �1,0,0� and �0,1,0� while the linkage channel-
like region remains in the �0,0,1� quasicrystallographic direc-tion as shown in Fig. 6�a� for c=0.7 and c=2.2. If kmax ischosen along the �1,1,1� direction the electron density func-tion is compressed in all three directions �depicted in Fig.6�b� for c=0.2 and c=3.0�. It is interesting to note that forthe same value of the electron wave vector k, the electronlocalization is stronger when k is along the �1,1,1� direction.Although not shown in the figures, the electron density func-tions of the exited states have more complicated shapes thanthose of the ground states. The equal density surfaces for theexcited states do not form closed contours inside the unit cellcubes, which indicate that the electrons are completely delo-calized and occupy the whole QDS.
The constant surfaces of the hole density functions ��nh�2
= cV0
for different exited states are shown in Figs. 7�a� and7�b�. Figure 7�a� shows the surface with c=2.0 for nh=12and k=0 for the quasicrystallographic direction �1,1,1�. Theequal density surfaces in this case are connected in manypoints. The holes with the densities indicated in the plots �orhigher densities� are localized inside of the unit cell of QDS.
( )a
( )b
FIG. 6. �Color online� Constant surface of the electron densityfunction for �a� c=0.7 �outside surface� and c=2.2 �inside surface�for the electron wave vector k=kmax�1,0 ,0�; and for �b� c=0.2 �out-side surface� and c=3.0 �inside surface� for k=kmax�1,1 ,1�. TheQDS dimensions are the same as in Fig. 5.
( )a
( )b
FIG. 7. �Color online� Constant surface of the hole density func-tion ��n=12
h �2= cd1d2d3
in the ordered QDS with the dimensions lx
= ly = lz=5 nm and hx=hy =hz=2 nm for �a� c=2.0 and the holewave vector k=0; and �b� c=1.7 for k=kmax�1,1 ,1� /2.
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Figure 7�b� presents the hole density function surface, whichcorresponds to the density ��n
h�2�1.7V0
at k=kmax�1,1 ,1� /2.One can see from this figure that the “open channels” for thehole transport through the ordered QDS do exist.
In order to quantitatively characterize the wave-functionpenetration in the neighboring unit cells of the ordered QDScells we calculated the dependence of the “delocalization”functions gn
e,h�k�= �B��n
e,h�r�� �dr� on the wave vector k forthe electrons �Fig. 8�a�� and holes �Fig. 8�b��. The values ofthe normalized functions gn
e,h�k� are given in percent. Theintegration in gn
e,h�k� is performed over the whole barriervolume �B between the QD. It follows from Fig. 8�a� thatthe function ge for the ground electron miniband reaches itsmaximum value at k=0, and decreases with increasing k forall directions of the electron motion. The electron wave func-tion for k=kmax�1,1 ,1� practically does not penetrate the bar-rier material and the electron is localized inside the QD. Adifferent situation is observed for the excited miniband n=3. For both quasicrystallographic directions �1,0,0� and�1,1,0� the confinement weakens with increasing wave vectork. For the �1,1,1� direction only the confinement enhancesslightly for the wave vector k close to kmax. The “delocaliza-tion” functions gn
h�k=0� for the hole states are smaller thanthose for the electrons states, and generally decrease withincreasing k. The stronger hole localization is explained bythe larger effective masses of the holes as compared to thoseof the electrons. The described properties of the electron andhole wave functions in QDS strongly influence the matrixelements of the electron �hole� interaction with photons andphonons.
V. OSCILLATOR STRENGTH OF THE OPTICALTRANSITIONS IN THE QUANTUM DOT SUPERLATTICES
The efficiency of the light absorption ability of a nano-structure can be illustrated, to some degree, by the oscillatorstrength of the optical transitions. For the calculation of theoscillator strength of the band-to-band �hole-electron� transi-tions we assumed that the electron momentum is equal to thehole momentum in absolute value and have the opposite di-rection, i.e., k�e=−k�h. The latter is a valid assumption due tothe small value of the photon momentum. The oscillatorstrength of the transition induced by the light polarized alongthe X axis, which leads to the generation of the electron-holepair, is given by
f�ne,nh� =Ep
Ee-hne,nh
= ����
V
��Re,ene,��r���1,Re,h
nh,� �r��
− �Im,ene,��r���1,Im,h
nh,� �r���dr�2
+ �V
��Re,ene,��r���1,Im,h
nh,� �r��
+ �Im,ene,��r���1,Re,h
nh,� �r���dr�2� , �11�
where Ep is the Kane energy, Ee-hne,nh is the energy of the
electron-hole pair with the quantum numbers ne, nh, �Re,ene,��r��
and �Im,ene,��r�� are the real and imaginary parts of the electron
wave function in the state with the quantum number ne,�Re,1,h
nh,� �r�� and �Im,1,hnh,� �r�� are the real and imaginary parts of
the hole wave-function component �1,hn,��r�� in the state with
the quantum number nh,� is the index of degeneracy. Wederived Eq. �11� using the formalism described in Ref. 32.
The dependence of the oscillator strengths on the photonenergy is shown in Figs. 9�a�–9�c� for the different param-eters of the quantum dots and various QDS quasicrystallo-graphic orientations: �1,0,0�, �1,1,0�, and �1,1,1�. Figure 9shows the oscillator strength in QDS with the dimensionslx=5 nm, ly =5.5 nm, lz=6 nm, and hx=hy =hz=2 nm for�1,0,0� orientation of QDS. The solid lines in this plot corre-spond to the transitions involving the twofold degenerateground electron state and 20 hole levels. The energy of thehigher electron states are considerably larger than the elec-tron ground-state energy and, therefore, do not affect theoscillator strength. The higher hole levels �n 20� do notinfluence the oscillator strength owing to the strong oscilla-tory nature of their wave functions. Figures 9�b� and 9�c�present the oscillator strength for two other orientations ofQDS. The electron-momentum independent oscillatorstrength can be obtained by averaging of the dependenciespresented in Figs. 9�a�–9�c� over all directions. Since thegeneral trends for the oscillator strength are similar for allthree quasicrystallographic directions, the averaged oscillatorstrength is expected to be very close to those depicted inFigs. 9�a�–9�c�.
The dashed lines in Figs. 9�a�–9�c� correspond to the os-cillator strength for the transitions between the ground-state
k/kmax(1,1,1)
(a)
g(
%)
nin
e
( ,0,0)k
( , , )k k k
( , ,0)k k
n =1h
n =6h
n =11h
n =17h
n =16h
( , , )k k k
(b)
g(
%)
nin
h
FIG. 8. Normalized “delocalization” function gne,h�k�
= �B��n
e,h�r�� �dr� as a function of the electron wave vector k for �a�the two lowest electron minibands �n=1, 2�; and �b� several holeminibands �m=1, 6, 11, 16, 17� in InAs/Ga0.5In0.5As QDS with thedimensions lx= ly = lz=5 nm and hx=hy =hz=2 nm.
CHARGE-CARRIER STATES AND LIGHT ABSORPTION IN… PHYSICAL REVIEW B 76, 125417 �2007�
125417-7
electrons and the eight lowest hole levels while the dottedlines correspond to the transitions between the ground-stateelectrons and the 12 hole levels with the quantum numbersnh=9–20. It is interesting to see from these plots that thehigher hole levels begin to participate in the transitions as theenergy exceeds �0.74 eV while the oscillator strength forthe transitions involving eight lower levels decrease to zerofor the energy �� 0.75 eV. Both the dashed and dottedcurves manifest a maximum value, i.e., the dashed curvesattain it at the photon energy of ��=0.65 eV while the dot-ted curves reach it at about ��=0.75 eV. The resulting os-cillator strength dependence on photon energy manifests ageneral monotonic decrease but remains rather high in a rela-tively wide interval pertinent to the infrared and visible lightabsorption. The latter suggest that such nanostructures can be
particularly efficient for the infrared photodetector applica-tions.
Another important application of QDS, which can greatlybenefit from the formation of the minibands and increasedoscillator strength is QD-based lasers. The Auger recombina-tion is the main reason for the decrease of the inverse chargepopulation in QD. The population inversion is a requiredcondition for generating the laser radiation.33 The higherdensity of QD density and formation of the minibands rein-force the effect of the stimulated emission and reduces theradiation relaxation time of the electron-hole pairs. This al-lows one to obtain strong laser radiation at the lower pumplevels.34 The miniband formation leads to the higher carriermobility, lower critical current density for electrical injectionand improved monochromaticity of the radiated light.
VI. CONCLUSIONS
We calculated the energy dispersion for the electrons andholes in the ordered quantum dot superlattices for the case ofInAs/Ga0.5In0.5As material system. From the obtained dis-persion we determined the oscillator strength of the opticaltransitions in such nanostructures. It has been shown that theenergy dispersion for the ground electron and hole states inthe quantum dot superlattices are nearly parabolic. The en-ergy of the electron excited states at zone center k=0 �size-quantized levels� is significantly larger than the potential bar-rier for the electrons and practically do not play any role inthe optical and kinetic processes. The energy spectra of thehole states obtained within the framework of the six-bandHamiltonian possess the features of the hybrid “heavy,”“light,” and “split-off” hole bands. The lowest energy levelsmanifest the weak energy dispersions similar to the “heavyholes” while the exited states display the strong dispersionsas the “light” and “split-off” holes. It has been establishedthat in the considered ordered QDS, with the selected geo-metrical parameters, heights of the potential barriers and val-ues of effective masses, both electrons and holes penetratethough the entire barrier layer thicknesses. Specifically, it hasbeen found via averaging of the charge densities that in thebarriers there are approximately 25% of the total electronand 18% of the total hole charge. The relevance of the de-veloped model for optimization of QDS for applications insolar cells, photodetectors, and lasers was also discussed.
ACKNOWLEDGMENTS
The work at UCR was supported, in part, through theAFOSR subcontract on the Optimized Quantum Dot Super-lattice Structures for High Efficiency Photovoltaic Cells andPhotodetectors. One of the authors �A.A.B.� is thankful to A.Fedoseyev and M. Turowski, CFD Research Corporation foruseful discussions. The authors also acknowledge the supportof the U.S. Civil Research and Development Foundation�CRDF�.
( , ,0)k k
nh=9-20nh=1-8
( ,0,0)k
nh=1-8nh=9-20
( , , )k k k
nh=1-8
nh=9-20
Photon Energy (eV)
Osc
illa
tor
Str
eng
th(a
rb.
un
.)O
scil
lato
rS
tren
gth
(arb
.u
n.)
Osc
illa
tor
Str
eng
th(a
rb.
un
.)
(a)
(b)
(c)
FIG. 9. Oscillator strength of the electron-hole optical transi-tions as a function of the photon energy in InAs/Ga0.5In0.5As QDSwith the dimensions �a� lx=5 nm, ly =5.5 nm, lz=6 nm, and hx=hy
=hz=2 nm for the �1,0,0� orientation; �b� lx=5 nm, ly =5 nm, lz
=6 nm, and hx=hy =hz=2 nm for �1,1,0�; and �c� lx= ly = lz=5 nm,lz=6 nm, and hx=hy =hz=2 nm for �1,1,1�.
NIKA et al. PHYSICAL REVIEW B 76, 125417 �2007�
125417-8
*On leave from the Department of Theoretical Physics, State Uni-versity of Moldova, Kishinev, Republic of Moldova
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